Homology of schemes and covariant motives

Abstract

In the present paper I will suggest a construction which assigns to the scheme S a tensor triangle category DM(S) and a covariant functor M from the category of schemes over S to DM(S), which satisfies the usual properties of homology theories. I hope that it gives us an appropriate theory of covariant mixed motives (except, that I have no idea how to prove the existence of the t-structure in DM(S)). This construction was inspired by topological analogs. The "homology theory of schemes" we obtain this way is related to the would-be homotopy theory of schemes in the same way as usual singular homologies of topological spaces are related to classical homotopy theory.